Epigraph form of an optimization problem Why is the optimization problem 
$\textrm{minimize} ~f_0(x)$ 
equivalent to 
$\textrm{minimize} ~~t \\ s.t. ~~ f_0(x)\leq t$
but not equivalent to the following?
$\textrm{minimize} ~~t \\ s.t. ~~ t=f_0(x)$
I found this example in Boyd's book and although he doesn't deny that the first and third minimization problems could be equivalent, it doesn't explicitly say so either? I'm wondering why he's using an inequality instead of an equality as a constraint?
 A: Suppose the original standard form of an optimization problem is:
$$\min~~f_0(\mathbf{x})\\ \text{s.t. } ~f_i(\mathbf{x}) \leq 0, i=1,...,m \\~~~~~~~ h_j(\mathbf{x}) = 0, j=1,...,p $$
in which  $f_i$ for  $i=1,...,m$ are inequality constraint functions and $h_j$ for $j=1,...,p$ are equality constraint functions. For a convex optimization problem, $f_0(\mathbf{x})$ and $f_i$ for $i=1,...,m$  are convex functions and $h_j$ for $j=1,...,p$ are affine functions. 
Now, for the equivalent epigraph representation of the original problem in standard form, we use the corresponding constraint in inequality form, we have:
$$\min ~~t \\ \text{s.t.} ~~ f_0(\mathbf{x}) - t\leq 0 \\ \qquad f_i(\mathbf{x}) \leq 0, i=1,...,m \\~~~~~~~ h_j(\mathbf{x}) = 0, j=1,...,p.$$ 
Assume the original problem is a convex optimization problem. To provide the original problem in epigraph standard representation, but preserving the problem to be in convex form, it needs to add an inequality constraint function $f_0(\mathbf{x}) - t$ which is convex in $(\mathbf{x},t)$; this inequality corresponds to the $\mathbf{epi} ~f_0$, here, is a convex set. So, the problem in the equivalent epigraph representation is still in a standard convex optimization problem form. Furthermore, for straightforward and meaningful analysis of a problem, also designing an efficient algorithm, different equivalent representation of a problem can be used. 
Note: $f$ is convex if and only if $\textbf{epi} ~f$ is convex set.
